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is smooth formal. When it is (for example when a Q-∆ version of the ∂-∂ Lemma holds) there is a weak-Frobenius man-ifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, a necessary and sufficient condition for L to be smooth formal is presented. The condition is simply stated: it(More)
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability(More)
We describe a step toward quantizing deformation theory. The L∞ operad is encoded in a Hochschild cocyle • 1 in a simple universal algebra (P, • 0). This Hochschild cocyle can be extended naturally to a star product ⋆ = • 0+ • 1+ 2 • 2+ · · ·. The algebraic structure encoded in ⋆ is the properad Ω(coF rob) which, conjecturally, controls a quantization of(More)
THE PROGRAM The Ph.D. Program in Mathematics provides students of high ability and strong preparation with an opportunity to begin study for the doctoral degree either immediately upon graduation from college or after completing some graduate work in the colleges of the City University or at other accredited institutions. Doctoral work in mathematics is(More)